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Theorem eunex 4203
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)
Assertion
Ref Expression
eunex  |-  ( E! x ph  ->  E. x  -.  ph )

Proof of Theorem eunex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dtru 4201 . . . . 5  |-  -.  A. x  x  =  y
2 alim 1545 . . . . 5  |-  ( A. x ( ph  ->  x  =  y )  -> 
( A. x ph  ->  A. x  x  =  y ) )
31, 2mtoi 169 . . . 4  |-  ( A. x ( ph  ->  x  =  y )  ->  -.  A. x ph )
43exlimiv 1666 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  -.  A. x ph )
54adantl 452 . 2  |-  ( ( E. x ph  /\  E. y A. x (
ph  ->  x  =  y ) )  ->  -.  A. x ph )
6 nfv 1605 . . 3  |-  F/ y
ph
76eu3 2169 . 2  |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
8 exnal 1561 . 2  |-  ( E. x  -.  ph  <->  -.  A. x ph )
95, 7, 83imtr4i 257 1  |-  ( E! x ph  ->  E. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143
This theorem is referenced by:  reusv2lem2  4536  unnt  24847  amosym1  24865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147
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